A zero sum differential game with correlated informations on the initial position. A case with a continuum of initial positions

We study a two player zero sum game where the initial position \begin{document}$ z_0 $\end{document} is not communicated to any player. The initial position is a function of a couple \begin{document}$ (x_0,y_0) $\end{document} where \begin{document}$ x_0 $\end{document} is communicated to player Ⅰ while \begin{document}$ y_0 $\end{document} is communicated to player Ⅱ. The couple \begin{document}$ (x_0,y_0) $\end{document} is chosen according to a probability measure \begin{document}$ dm(x,y) = h(x,y) d\mu(x) d\nu(y) $\end{document} . We show that the game has a value and, under additional regularity assumptions, that the value is a solution of Hamilton Jacobi Isaacs equation in a dual sense.